AES branch number?

Question

Mathematics

By definition branch number Taken from here (7.3.1)
Where W(a) is the weight of the vector i.e. number of nonzero vector components

AES uses a predefined matrix in the MixColumns operation . It is necessary to prove that for it (the matrix) branch number = 5.
Questions:
1. In the same article (7.3.1) it is said that

the output can have at most 4 active bytes

and

Hence, the upper bound for the branch number is 5

It turns out that W(F(a)) maximum can be equal to 4 (why?) and W(a) = 1. Why W(a) = 1, because the number of non-zero components can be greater than 1? Or is this where you should pay attention to min()?
2. How to calculate W(F(a)), for each